Finding eigenvalue and normalized eigenstate of a hamiltonian

The system described by the Hamiltonian [tex] H_0[/tex] has just two orthogonal energy eigenstates, |1> and |2> , with

<1|1>=1 , <1|2> =0 and <2|2>=1 . The two eignestates have the same eigenvalue , E_0:

H_0|i>=E_0|i>, for i=1 and 2.

Now suppose the Hamiltonian for the system is changed by the addition of the term V, given H=H_0+V.

The matrix elements of V are

<1|V|1> =0 , <1|V|2>=V_12, <2|V|2>=0.

a) Find the eigenvalues of the new Hamiltonian, H , in terms of the quanties above

b) Find the normalized eigenstates of H in terms of |1> , |2> and the other given expressions.

2. Relevant equations

3. The attempt at a solution
a) I don't know how to begin this problem but I guess I will start by plugging in the values for H_0 and V: H=H_0+V=<1|1>+<1|V|1>=1+0=1 for the first value of H_0 and the first value of V. don't you find the eigen value by writing out this expression: det(H-I*lambda)=0? I have kno w idea.

You would do the hamiltonian-lambda*I and you would find the determinate of det(hamiltonian-lambda*I )=0 to find the values of lambda right? then I would normalized the hamiltonian?

Yes then no. You are correct in finding your eigenvalues ([itex]\lambda[/itex]), but you are not asked to normalize the Hamiltonian, you are asked to find the normalized eigenstates (which are the same thing as eigenvectors). If you are unfamiliar with this process, then I would recommend checking out http://www.tmt.ugal.ro/crios/Support/ANPT/Curs/math/s3/s3eign/s3eign.html" [Broken]

This is correct. I will note that there will be a condition involved on [itex]V_{12}[/itex] and [itex]V_{21}[/itex] in finding your eigenvalues. When finding [itex]\lambda[/itex], remember that your eigenvalues must be real.